Weakly nonlinear theory is used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, i.e. Lapwood convection in a saturated porous layer heated from below. Two particular aspects of the problem are focused upon: (i) the effect of thermal imperfections on the stability characteristics of steady rolls near onset; and (ii) the evolution of unstable rolls.For Rayleigh-Bénard convection it is well known (see Busse and co-workers 1974, 1979, 1986) that the stability of steady two-dimensional rolls near onset is limited by the presence of cross-roll, zigzag and sideband disturbances; this is shown to be true also in Lapwood convection. We further determine the modifications to the stability boundaries when small-amplitude imperfections in the boundary temperatures are present. In practice imperfections would usually consist of broadband thermal noise, but it is the Fourier component with wavenumber close to the critical wavenumber for the perfect problem (i.e. in the absence of imperfections) which, when present, has the greatest effect due to resonant forcing. This particular case is the sole concern of the present paper; other resonances are considered in a complementary study (Rees & Riley 1989).For the case when the modulations on the upper and lower boundaries are in phase, asymptotic analysis and a spectral method are used to determine the stability of roll solutions and to calculate the evolution of the unstable flows. It is shown that steady rolls with spatially deformed axes or spatially varying wavenumbers evolve. The evolution of the flow that is unstable to sideband disturbances is also calculated when the modulations are π out of phase. Again rolls with a spatially varying wavenumber result.
Weakly nonlinear stability analysis of triple diffusive convection in a Maxwell fluid saturated porous layer
